On Korn’s inequalities on a surface

Abstract

Korn’s inequalities on a surface constitute the keystone for establishing the existence and uniqueness of solutions to various linearly elastic shell problems. As a rule, they are, however, somewhat delicate to establish. After briefly reviewing how such Korn inequalities are classically established, we show that they can be given simpler and more direct proofs in some important special cases, without any recourse to J. L. Lions lemma; besides, some of these inequalities hold on open sets that are only assumed to be bounded. In particular, we establish a new “identity for vector fields defined on a surface”. This identity is then used for establishing new Korn’s inequalities on a surface, whose novelty is that only the trace of the linearized change of curvature tensor appears in their right-hand side.